.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/fields/electrodynamics/plot_dielectric_slab_propagation.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_fields_electrodynamics_plot_dielectric_slab_propagation.py: A wave crossing a dielectric slab ======================================= Kane Yee's staggered-grid update handles a wave crossing a material interface with no change to its core leapfrog rule -- only the permittivity map ``eps_r`` fed into it changes: the TMz-mode Maxwell curl equations solved at every grid cell, .. math:: \frac{\partial H_x}{\partial t} = -\frac{1}{\mu_0}\frac{\partial E_z}{\partial y}, \qquad \frac{\partial H_y}{\partial t} = \frac{1}{\mu_0}\frac{\partial E_z}{\partial x}, \qquad \frac{\partial E_z}{\partial t} = \frac{1}{\varepsilon_0\varepsilon_r(x,y)}\left(\frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y}\right), simply pick up a spatially varying :math:`\varepsilon_r(x,y)` instead of the vacuum value 1. :func:`~physicskit.fields.electrodynamics.dielectric_slab` builds exactly such a map: vacuum everywhere except a slab (spanning grid columns ``i_start`` to ``i_end``) of elevated permittivity ``eps_r_slab``, so a broad Gaussian plane-wave pulse launched toward it partially reflects and partially refracts/transmits at each interface, exactly as an ordinary glass slab does to light. .. GENERATED FROM PYTHON SOURCE LINES 25-31 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.fields import EPS0, MU0, animate_field_2d, courant_limit_2d, dielectric_slab, fdtd_2d_tmz_evolve .. GENERATED FROM PYTHON SOURCE LINES 32-34 A broad plane-wave pulse launched toward a dielectric slab ------------------------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 34-50 .. code-block:: Python Nx, Ny, dx, dy = 220, 60, 1e-3, 1e-3 dt = 0.5 * courant_limit_2d(dx, dy) eta0 = np.sqrt(MU0 / EPS0) x0, sigma = 40, 8 i = np.arange(Nx) Ez0 = np.tile(np.exp(-((i - x0) ** 2) / (2 * sigma**2)), (Ny, 1)).T ih = i + 0.5 Hy0 = np.tile(np.exp(-((ih - x0) ** 2) / (2 * sigma**2)) / eta0, (Ny, 1)).T Hx0 = np.zeros((Nx, Ny)) i_start, i_end, eps_r_slab = 110, 160, 4.0 eps_r = dielectric_slab((Nx, Ny), i_start=i_start, i_end=i_end, eps_r_slab=eps_r_slab) mu_r = np.ones((Nx, Ny)) .. GENERATED FROM PYTHON SOURCE LINES 51-53 Leapfrog the staggered E/H update forward, recording snapshots ------------------------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 53-57 .. code-block:: Python steps = 420 frames, times = fdtd_2d_tmz_evolve(Ez0, Hx0, Hy0, eps_r, mu_r, steps=steps, dt=dt, dx=dx, dy=dy, snapshot_stride=6) .. GENERATED FROM PYTHON SOURCE LINES 58-62 By the final frame, part of the pulse has reflected off the slab's entrance interface (visible back in the vacuum region), and part has transmitted through and out the far side, at a speed slowed by :math:`1/\sqrt{\varepsilon_r}` while inside the slab. .. GENERATED FROM PYTHON SOURCE LINES 62-70 .. code-block:: Python x = np.arange(Nx) * dx y = np.arange(Ny) * dy X, Y = np.meshgrid(x, y, indexing="ij") anim = animate_field_2d(X, Y, frames, times) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 71-75 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("dielectric_slab.gif", writer="pillow", fps=20) .. GENERATED FROM PYTHON SOURCE LINES 75-83 .. code-block:: Python reflected_region = frames[-1, : i_start - 20, Ny // 2] transmitted_region = frames[-1, i_end + 5 :, Ny // 2] fresnel_r = abs((1.0 - np.sqrt(eps_r_slab)) / (1.0 + np.sqrt(eps_r_slab))) print(f"peak |Ez| reflected back into vacuum: {np.max(np.abs(reflected_region)):.4f}") print(f"peak |Ez| transmitted past the slab: {np.max(np.abs(transmitted_region)):.4f}") print(f"normal-incidence Fresnel |r| for eps_r={eps_r_slab}: {fresnel_r:.4f} (single-interface reference)") .. rst-class:: sphx-glr-script-out .. code-block:: none peak |Ez| reflected back into vacuum: 0.7193 peak |Ez| transmitted past the slab: 0.0002 normal-incidence Fresnel |r| for eps_r=4.0: 0.3333 (single-interface reference) .. GENERATED FROM PYTHON SOURCE LINES 84-95 A space-time diagram along the beam axis -------------------------------------------- The animation shows the full 2D field but only one frame at a time; stacking the centerline (``y = Ny//2``) of every already-recorded snapshot into a single image shows the whole history at once: the incident pulse's constant-slope (speed :math:`c_0`) ridge, a reflected ridge peeling back off the slab's entrance interface, and a visibly shallower-sloped ridge -- moving at :math:`c_0/\sqrt{\varepsilon_r}`, half the vacuum speed for :math:`\varepsilon_r=4` -- still partway across the slab by the final recorded frame, not yet having reached the exit interface at this run's ``steps``. .. GENERATED FROM PYTHON SOURCE LINES 95-107 .. code-block:: Python centerline = frames[:, :, Ny // 2] fig2, ax2 = plt.subplots() extent = (x.min(), x.max(), times.min(), times.max()) im = ax2.imshow(centerline, extent=extent, origin="lower", aspect="auto", cmap="RdBu_r", vmin=-1, vmax=1) ax2.axvspan(i_start * dx, i_end * dx, color="gray", alpha=0.25, label=f"slab (eps_r={eps_r_slab})") fig2.colorbar(im, ax=ax2, label="Ez") ax2.set_xlabel("x") ax2.set_ylabel("t") ax2.set_title("Space-time diagram along the centerline") ax2.legend(loc="upper left") fig2.tight_layout() .. image-sg:: /api/gallery/fields/electrodynamics/images/sphx_glr_plot_dielectric_slab_propagation_002.png :alt: Space-time diagram along the centerline :srcset: /api/gallery/fields/electrodynamics/images/sphx_glr_plot_dielectric_slab_propagation_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 9.193 seconds) .. _sphx_glr_download_api_gallery_fields_electrodynamics_plot_dielectric_slab_propagation.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_dielectric_slab_propagation.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_dielectric_slab_propagation.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_dielectric_slab_propagation.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_